Number Theory Seminar

Speaker: Tristan Phillips (University of Arizona)

Title: Counting Points on Modular Curves over Global Fields

Abstract: Let E be an elliptic curve over a number field K. The Mordell-Weil Theorem states that the set of rational points E(K) of E forms a finitely generated abelian group. In particular, we may write E(K)≅ E(K)_tors⊕ ℤr, where E(K)_tors is a finite torsion group, called the torsion subgroup of E, and r is a non-negative integer, called the rank of E. In this talk I will discuss some results regarding how frequently elliptic curves with a prescribed torsion subgroup occur, and how one can bound the average analytic rank of elliptic curves over arbitrary number fields. One of the main ideas behind these results is to use methods from Diophantine geometry to count points of bounded height on modular curves. If time permits, I will discuss ongoing work on a function field analog of some of these results, which have applications to the arithmetic statistics of Drinfeld modules.

The talk will take place on Zoom. If you are interested in attending and not on the Number Theory Seminar mailing list, please contact the organizer to obtain the link